a b s t r a c tIn this paper, we consider the problem of manifold approximation with affine subspaces. Our objective is to discover a set of low dimensional affine subspaces that represent manifold data accurately while preserving the manifold's structure. For this purpose, we employ a greedy technique that partitions manifold samples into groups, which are approximated by low dimensional subspaces. We start by considering each manifold sample as a different group and we use the difference of local tangents to determine appropriate group mergings. We repeat this procedure until we reach the desired number of sample groups. The best low dimensional affine subspaces corresponding to the final groups constitute our approximate manifold representation. Our experiments verify the effectiveness of the proposed scheme and show its superior performance compared to state-of-the-art methods for manifold approximation.

Sparsity has been one of the major drives in signal processing in the last decade. Structured sparsity has also lately emerged as a way to enrich signal priors towards more meaningful and accurate representations. In this paper we propose a new structured sparsity signal model that allows for the decomposition of signals into structured molecules. We define the molecules to be linear combinations of atoms in a dictionary and we create a decomposition scheme that allows for their identification in noisy signals while being robust to small errors in the internal molecule structure. We show the effectiveness of our scheme for recovering and identifying corrupted or occluded signals on both synthetic and real data.

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Sparsity-based models have proven to be very effective in most image processing applications. The notion of sparsity has recently been extended to structured sparsity models where not only the number of components but also their support is important. This paper goes one step further and proposes a new model where signals are composed of a small number of molecules, which are each linear combinations of a few elementary functions in a dictionary. Our model takes into account the energy on the signal components in addition to their support. We study our prior in detail and propose a novel algorithm for sparse coding that permits the appearance of signal dependent versions of the molecules. Our experiments prove the benefits of the new image model in various restoration tasks and confirm the effectiveness of priors that extend sparsity in flexible ways especially in case of inverse problems with low quality data.